A Tour of NTL: Summary of NTL's Main Modules

NTL consists of a number of software modules. Generally speaking, for each module foo, there is

• a header file <NTL/foo.h>, found in subdirectory include,
• a documentation file foo.txt, found in subdirectory doc, and
• a source file foo.c, found in subdirectory src.

Note that all of the header files for NTL modules include the header file <NTL/tools.h>, and this header file includes the standard headers

• <stdlib.h>,
• <math.h>, and
• <iostream.h>;

Also note that in ISO mode, <NTL/tools.h> instead includes the standard header files

• <cstdlib>,
• <cmath>, and
• <iostream>;

The documentation file takes the form of a header file, but stripped of implementation details and declarations of some of the more esoteric routines and data structures, and it contains more complete and usually clearer documentation than in the header file.

There is a plethora of conversion routines. These are not documented in any of the individual documentation files, but rather, they are all briefly summarized in conversions.txt.

The following is a summary of the main NTL modules. The corresponding documentation file can be obtained by clicking on the module name.

GF2	class GF2: integers mod 2
GF2X	class GF2X: polynomials over GF(2) (much more efficient than using zz_pX with p=2); includes routines for GCDs and minimal polynomials
GF2XFactoring	routines for factoring polynomials over GF(2); also includes routines for testing for and constructing irreducible polynomials
GF2XVec	class GF2XVec: fixed-length vectors of fixed-length GF2Xs; less flexible, but more efficient than vec_GF2X
GF2E	class GF2E: polynomial extension field/ring over GF(2), implemented as GF(2)[X]/(P).
GF2EX	class GF2EX class GF2EX: polynomials over GF2E; includes routines for modular polynomials arithmetic, modular composition, minimal and characteristic polynomials, and interpolation.
GF2EXFactoring	routines for factoring polynomials over GF2E; also includes routines for testing for and constructing irreducible polynomials
HNF	routines for computing the Hermite Normal Form of a lattice
LLL	routines for performing lattice basis reduction, including very fast and robust implementations of the Schnorr-Euchner LLL and Block Korkin Zolotarev reduction algorithm, as well as an integer-only reduction algorithm. Also, there are routines here for computing the kernel and image of an integer matrix, as well as finding integer solutions to linear systems of equations over the integers.
RR	class RR: arbitrary-precision floating point numbers.
ZZ	class ZZ: arbitrary length integers; includes routines for GCDs, Jacobi symbols, modular arithmetic, and primality testing; also includes small prime generation routines and in-line routines for single-precision modular arithmetic
ZZVec	class ZZVec: fixed-length vectors of fixed-length ZZs; less flexible, but more efficient than vec_ZZ
ZZX	class ZZX: polynomials over ZZ; includes routines for GCDs, minimal and characteristic polynomials, norms and traces
ZZXFactoring	routines for factoring univariate polynomials over ZZ
ZZ_p	class ZZ_p: integers mod p
ZZ_pE	class ZZ_pE: ring/field extension of ZZ_p
ZZ_pEX	class ZZ_pEX: polynomials over ZZ_pE; includes routines for modular polynomials arithmetic, modular composition, minimal and characteristic polynomials, and interpolation.
ZZ_pEXFactoring	routines for factoring polynomials over ZZ_pE; also includes routines for testing for and constructing irreducible polynomials
ZZ_pX	class ZZ_pX: polynomials over ZZ_p; includes routines for modular polynomials arithmetic, modular composition, minimal and characteristic polynomials, and interpolation.
ZZ_pXFactoring	routines for factoring polynomials over ZZ_p; also includes routines for testing for and constructing irreducible polynomials
lzz_p	class zz_p: integers mod p, where p is single-precision
lzz_pE	class zz_pE: ring/field extension of zz_p
lzz_pEX	class zz_pEX: polynomials over zz_pE; provides the same functionality as class ZZ_pEX, but for single-precision p
lzz_pEXFactoring	routines for factoring polynomials over zz_pE; provides the same functionality as class ZZ_pEX, but for single-precision p
lzz_pX	class zz_pX: polynomials over zz_p; provides the same functionality as class ZZ_pX, but for single-precision p
lzz_pXFactoring	routines for factoring polynomials over zz_p; provides the same functionality as class ZZ_pX, but for single-precision p
matrix	template-like macros for dynamic-size 2-dimensional arrays
mat_GF2	class mat_GF2: matrices over GF(2); includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
mat_GF2E	class mat_GF2E: matrices over GF2E; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
mat_RR	class mat_RR: matrices over RR; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, and solving nonsingular systems of linear equations.
mat_ZZ	class mat_ZZ: matrices over ZZ; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, and solving nonsingular systems of linear equations. See also the LLL module for additional routines.
mat_ZZ_p	class mat_ZZ_p: matrices over ZZ_p; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
mat_ZZ_pE	class mat_ZZ_pE: matrices over ZZ_pE; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
mat_lzz_p	class mat_zz_p: matrices over zz_p; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
mat_lzz_pE	class mat_zz_pE: matrices over zz_pE; includes basic matrix arithmetic operations, including determinant calculation, matrix inversion, solving nonsingular systems of linear equations, and Gaussian elimination
mat_poly_ZZ	routine for computing the characteristic polynomial of a mat_ZZ
mat_poly_ZZ_p	routine for computing the characteristic polynomial of a mat_ZZ_p
mat_poly_lzz_p	routine for computing the characteristic polynomial of a mat_zz_p
pair	template-like macros for pairs
quad_float	class quad_float: quadruple-precision floating point numbers.
tools	some basic types and utility routines, including the timing function GetTime(), and several overloaded versions of min() and max()
vector	template-like macros for dynamic-size vectors
vec_GF2	class vec_GF2: vectors over GF(2), with arithmetic
vec_GF2E	class vec_GF2E: vectors over GF2E, with arithmetic
vec_RR	class vec_RR: vectors over RR, with arithmetic
vec_ZZ	class vec_ZZ: vectors over ZZ, with arithmetic
vec_ZZ_p	class vec_ZZ_p: vectors over ZZ_p, with arithmetic
vec_ZZ_pE	class vec_ZZ_pE: vectors over ZZ_pE, with arithmetic
vec_lzz_p	class vec_zz_p: vectors over zz_p, with arithmetic
vec_lzz_pE	class vec_zz_pE: vectors over zz_pE, with arithmetic
version	macros defining the NTL version number
xdouble	class xdouble: double-precision floating point numbers with extended exponent range.

Some other types

In addition to the above, other generic vectors are declared, not explicitly documented elsewhere:

• vec_GF2XVec
• vec_ZZVec
• vec_double
• vec_long
• vec_quad_float
• vec_ulong
• vec_vec_GF2
• vec_vec_GF2E
• vec_vec_RR
• vec_vec_ZZ
• vec_vec_ZZ_p
• vec_vec_ZZ_pE
• vec_vec_long
• vec_vec_lzz_p
• vec_vec_lzz_pE
• vec_vec_ulong
• vec_xdouble

These decalarations are found in ".h" files with corresponding names. No additional functionality is provided.

All of the header files for polynomial classes ZZ_pX, ZZX, etc., declare generic vectors vec_ZZ_pX, vec_ZZX, etc.

There are also a number of generic pair classes defined, not explicitly documented elsewhere:

• pair_GF2EX_long
• pair_GF2X_long
• pair_ZZX_long
• pair_ZZ_pEX_long
• pair_ZZ_pX_long
• pair_lzz_pEX_long
• pair_lzz_pX_long

These decalarations are found in ".h" files with corresponding names. These files also declare corresponding generic vector types vec_pair_GF2EX_long, etc. No additional functionality is provided.